Mr. McMath,
The soldier's Compiled Military Service Record will possibly have the hospital's name. There were several in the Chattanooga area at time. Burial location would be problematic. There are about 144 soldiers buried in the Silverdale Confederate Cemetery. These soldiers died of disease in the summer of 1862. No know list of these soldiers is known to exist.
http://www.utc.edu/Academic/Communication/ConfCem/ConfederateCem.html
A good reference for you to learn about the hospital system and the ones in Chattanooga during that time period would be "Confederate Hospitals on the Move: Samuel H. Stout and the Army of Tennessee." by Glenna R. Schroeder-Lein.
I hope this helps.
Respectfully,
Gerald D. Hodge, Jr.
War Between the States Historian
Historian: 39th Georgia Volunteer Infantry Regiment
http://39thGAVOLINFRGT.homestead.com/39thHomepage.html